Fuzzy Propositional Logic Research Articles

Unified theory of integral true degrees in NM theory

Based on the NM propositional logic system of nilpotent minimum logic, the concept of the integral truth degree of the formula is firstly proposed by integrating the function induced by the formula, and the MP rule and HS rule of the integral truth degree are proved by means of the integral invariance. Secondly, in NM propositional logic the integral similarity and integral pseudo-distance are introduced into the set of general formulas of the system, and some good properties about the similarity and pseudo-distance are proved. Finally, in terms of the concept of divergence degrees and diameter, a new membership function for reflecting the consistency degrees of theories in NM propositional fuzzy logic is proposed, which is proved that of inconsistent theories are equal to 0 and that of completely consistent theories are equal to 1.

A lattice-theoretical approach to extensions of filters in algebras of substructural logic

Commutative bounded integral residuated lattices (residutaed lattices, in short) form a large class of algebras containing algebras which are algebraic counterparts of certain propositional fuzzy logics. The paper deals with the so-called extended filters of filters of residuated lattices. It is used the fact that the extended filters of filters associated with subsets coincide with those associated ones with corresponding filters. This makes it possible to investigate the set of all extended filters of residuated lattices within the Heyting algebras of their filters by means of the structural methods of the theory of such algebras.

Logical Characterizations of Fuzzy Simulations

We provide and prove logical characterizations of fuzzy simulations between fuzzy Kripke models that use a general t-norm-based semantics. We also extend the results for fuzzy Kripke models defined over a general residuated lattice. Our logical characterizations of fuzzy simulations are formulated w.r.t. positive existential fragments of fuzzy propositional dynamic logic without tests and concern fuzzy preservation of positive existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.

Logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices

There are two kinds of bisimulation, namely crisp and fuzzy, between fuzzy structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models and fuzzy interpretations in description logics. Fuzzy bisimulations between fuzzy automata over a complete residuated lattice have been introduced by Ćirić et al. in 2012. Logical characterizations of fuzzy bisimulations between fuzzy Kripke models (respectively, fuzzy interpretations in description logics) over the residuated lattice [0,1] with the Gödel t-norm have been provided by Fan in 2015 (respectively, Nguyen et al. in 2020). There was the lack of logical characterizations of fuzzy bisimulations between fuzzy graph-based structures over a general residuated lattice, as well as over the residuated lattice [0,1] with the Łukasiewicz or product t-norm. In this article, we provide and prove logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices. The considered logics are the fuzzy propositional dynamic logic and its fragments. Our logical characterizations concern invariance of formulas under fuzzy bisimulations and the Hennessy-Milner property of fuzzy bisimulations. They can be reformulated for other fuzzy structures such as fuzzy labeled transition systems and fuzzy interpretations in description logics.

ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem

Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO.

Characterizing fuzzy simulations for fuzzy labeled transition systems in fuzzy propositional dynamic logic

We formulate and prove logical characterizations of fuzzy simulations for fuzzy labeled transition systems under the Gödel semantics. Apart from the usual setting, we consider also the setting that allows graded modal operators, which are used to talk about the number of successors with a certain property of the current state. Our characterizations are formulated w.r.t. the existential fragment of fuzzy propositional dynamic logic and its extension with graded modal operators. The results concern fuzzy preservation of existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.

A novel plausible reasoning based on intuitionistic fuzzy propositional logic and its application in decision making

Automatic reasoning based on propositional logic is considered as an important tool in machine learning, and intuitionistic fuzzy sets have turned out to deal with vague and uncertain information effectively in real world. In this paper, a novel plausible reasoning based on intuitionistic fuzzy propositional logic is proposed. On the basis of it, the categories of intuitionistic fuzzy logic proposition (IFLP) formula are discussed both considering the true degree and the false degree at the same time. Some basic operational laws and inference rules of IFLPs on the basis of closely-reasoned scientific proofs are put out. Then, we develop two classification methods of IFLPs, i.e., truth table and figure of equivalence, respectively. After that, the reasoning theory of IFLPs is introduced and three reasoning methods are further established including direct proof method, additional premise proof method and reduction to absurdity method, respectively. Finally, a case study about strategy initiatives of HBIS GROUP on Supply-side Structural Reform is presented and some discussions are provided to validate the proposed methods. As a result, the proposed methods based on the plausible reasoning offer sound structure and can improve the efficiency of logic programming.

(α, β)-Ordered linear resolution of intuitionistic fuzzy propositional logic

This paper presents an approach for (α, β)-resolution-based automated reasoning in intuitionistic fuzzy propositional logic (IFL). In IFL systems, every IFL formula with both a truth degree and a falsity degree at the same time can be transformed into IFL generalized clause based on some equivalent properties in a finite time. With assumed thresholds (α, β), we judge whether each IFL literal of IFL generalized clauses is (α, β)-satisfiable or not. According to the IFL literal’s satisfiability, the IFL literals can be divided into two classes, i.e. (α, β)-similar literals and (α, β)-complementary literals. We use the structure of the linear resolution which only has a center line and a side line to restrict the resolution clauses in every resolution procedure. IFL framed literals, recording information of resolved literals instead of storing them in a memory, can be used to reduce the number of possible resolutions and improve the efficiency of the resolution. The empty clause can be obtained with layer resolution where the threshold (α, β) can be changed. The soundness and completeness of (α, β)-ordered linear resolution are proved and an illustrative example is given to verify the (α, β)-ordered linear resolution algorithm and demonstrate its effectiveness.

Derivations on algebras of a non-commutative generalization of the Łukasiewicz logic

GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic semantics of the non-commutative Łukasiewicz infinite valued propositional fuzzy logic. In the paper, derivations on GMV-algebras (which are formally introduced in the same manner as derivations on rings) are investigated. A complete description of all derivations on any GMV-algebra is given.

On lifting quasi-filters and strong lifting quasi-filters in MV-algebras

A lifting inference rule is a many-valued inference rule that manipulates with evaluated formulas, it has played an important role in Pavelka-style fuzzy propositional logic. In this paper, we consider the algebraic characterization of Pavelka-style fuzzy logic deductive system based on lifting rul es in MV-algebras from the algebraic semantical point of view, which can be a tool to analyze the provability in fuzzy logic with evaluated syntax based on the lifting inference rules. We introduce the notions of lifting quasi-filter and strong lifting quasi-filter in MV-algebras, they are the algebraic abstractions of the sets of provable formulae which are closed with respect to lifting inference rules in the corresponding formal logic system, and some important properties are presented. We also characterize two kinds of extended lifting quasi-filters and several kinds of generated lifting quasi-filters, and prove that the class F α (L) of all lifting quasi-filters with respect to α is an MV-algebra, the class Fs α (L) of all strong lifting quasi-filters with respect to α is a complete Heyting algebra.

Intuitionistic fuzzy logics as tools for evaluation of Data Mining processes

The Intuitionistic Fuzzy Sets (IFSs), proposed in 1983, are extensions of fuzzy sets. Some years after their introduction, sequentially, intuitionistic fuzzy propositional logic, intuitionistic fuzzy predicate logic, intuitionistic fuzzy modal logic and intuitionistic fuzzy temporal logic have been introduced, presented here shortly. During the last 25years, different intuitionistic fuzzy tools have been used for evaluation of objects from the area of the Artificial Intelligence, as expert systems (having, e.g. facts and rules, with intuitionistic fuzzy degrees of validity and non-validity), decision making processes (having, e.g. intuitionistic fuzzy estimations of the criteria), neural networks, pattern recognitions, metaheuristic algorithms, etc. Short review of these legs of research is offered, with some concrete ideas of possible new directions of study. On this basis, a non-formal discussion is raised on the benefits of applying various elements of intuitionistic fuzzy logics as tools for evaluation of Data Mining processes.

Semantics of Propositional Fuzzy Modal Logic with Evaluated Syntax and its Application to Fuzzy Decision Implications

AbstractThis paper deals with propositional fuzzy modal logic with evaluated syntax based on MV-algebras. We focus on its semantical theory from the viewpoint of Pavelka's graded semantics of propositional fuzzy logic, investigate the L-tautologies based on different Kripke frames. We also define the notion of L-semantical consequence operation, its some basic properties are obtained. Finally, this paper considers the fuzzy decision implications in propositional fuzzy modal logic with evaluated syntax based on MV-algebras, and presents a kind of semantical characteristics of fuzzy decision implications. Moreover, we introduce the notions of possible and necessary fuzzy decision implication, and their semantical characteristics are presented as well.

An algebraic study of Peterson’s Intermediate Syllogisms

Peterson's Intermediate Syllogisms, generalizing Aristotelian syllogisms by intermediate quantifiers `Many', `Most' and `Almost all', are studied. It is demonstrated that, by associating certain values V, W and U on standard ?ukasiewicz MV-algebra with the first and second premise and the conclusion, respectively, the validity of a corresponding intermediate syllogism is determined by a simple MV-algebra (in-)equation. Possible conservative extensions of Peterson's system are discussed. Finally it is shown that Peterson's bivalued intermediate syllogisms can be viewed as fuzzy theories in Pavelka's fuzzy propositional logic, i.e. a fuzzy version of Peterson's Intermediate Syllogisms is introduced.

Δ-core Fuzzy Logics with Propositional Quantifiers, Quantifier Elimination and Uniform Craig Interpolation

In this paper we investigate the connections between quantifier elimination, decidability and Uniform Craig Interpolation in Δ-core fuzzy logics added with propositional quantifiers. As a consequence, we are able to prove that several propositional fuzzy logics have a conservative extension which is a Δ-core fuzzy logic and has Uniform Craig Interpolation.

Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

Complex fuzzy sets, classes, and logic have an important role in applications, such as prediction of periodic events and advanced control systems, where several fuzzy variables interact with each other in a multifaceted way that cannot be represented effectively via simple fuzzy operations such as union, intersection, complement, negation, conjunction and disjunction. The initial formulation of these terms stems from the definition of complex fuzzy grade of membership. The problem, however, with these definitions are twofold: 1) the complex fuzzy membership is limited to polar representation with only one fuzzy component. 2) The definition is based on grade of membership and is lacking the rigor of axiomatic formulation. A new interpretation of complex fuzzy membership enables polar and Cartesian representation of the membership function where the two function components carry uncertain information. Moreover, the new interpretation is used to define complex fuzzy classes and develop an axiomatic based theory of complex propositional fuzzy logic. Additionally, the generalization of the theory to multidimensional fuzzy grades of membership has been demonstrated. In this paper we propose an axiomatic framework for first order predicate complex fuzzy logic and use this framework for axiomatic definition of complex fuzzy classes. We use these rigorous definitions to exemplify inference in complex economic systems. The new framework overcomes the main limitations of current theory and provides several advantages. First, the derivation of the new theory is based on axiomatic approach and does not assume the existence of complex fuzzy sets or complex fuzzy classes. Second, the new form significantly improves the expressive power and inference capability of complex fuzzy logic and class theory. The paper surveys the current state of complex fuzzy sets, complex fuzzy classes, and complex fuzzy logic; and provides an axiomatic basis for first order predicate complex fuzzy logic and complex class theory.

Some properties of fuzzy reasoning in propositional fuzzy logic systems

In order to analyze the logical foundation of fuzzy reasoning, this paper first introduces the concept of generalized roots of theories in Łukasiewicz propositional fuzzy logic Łuk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic Π, and nilpotent minimum logic NM (the R 0-propositional fuzzy logic L ∗ ). Next, it is proved that all consequences of a theory Γ, named D( Γ), are completely determined by its generalized root whenever Γ has a generalized root. Moreover, it is proved that every finite theory Γ has a generalized root, which can be expressed by a specific formula. Finally, we demonstrate the existence of a non-fuzzy version of Fuzzy Modus Ponens (FMP) in Łuk, Göd, Π and NM ( L ∗ ) , and we provide its numerical version as a new algorithm for solving FMP.

Consistency Degrees of Theories in Łukasiewicz Fuzzy andn-Valued Propositional Logic Systems

By means of theory of truth degrees of formulas, according to deduction theorems and completeness theorems, the new concepts of consistency degrees and polar index for general theories in Łukasewicz fuzzy andn-valued propositional logic systems are introduced. Moreover, sufficient and necessary conditions for a theoryΓto be consistent, inconsistent, and fully divergent are obtained. Finally, some important properties of truth degrees of formulas are proposed.

On fuzzy modal logics [formula omitted

The fuzzy variant S 5 ( C ) of the well-known modal logic S5 is studied, C being a recursively axiomatized fuzzy propositional logic extending the basic fuzzy logic BL. Three kinds of Kripke models are introduced and corresponding deductive systems are found.

Fuzzy logics with an additional involutive negation

This paper surveys the present state of knowledge on propositional fuzzy logics extending SBL with an additional involutive negation. The involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms.

A Formal Model of Ontology for Handling Fuzzy Membership and Typicality of Instances

Ontology has become increasingly important in facilitating information exchange, particularly in the context of the Semantic Web. Currently, most existing ontology models can only specify concepts as crisp sets. However, concepts that are without clear boundaries or are vague in meanings are abundant. Existing ontology models are therefore unable to cope with many real cases effectively. In addition, with respect to a certain category, certain objects can be considered as more representative or typical, which are explained by cognitive psychologists using the Prototype Theory of concepts. Based on this theory, we propose a formal model for fuzzy ontologies. This model is equipped with likeliness, the extent to which an object is considered as an instance of a concept, and typicality, the representativeness of an object in a concept. This model enables ontologies to model concepts and bring the results of reasoning closer to human thinking. Our work is based on an in-depth investigation of the limitations of existing models and findings in cognitive psychology. The nature and differences between likeliness and typicality are also thoroughly discussed. In addition, we present a logic for the ontology model which is based on fuzzy propositional modal logic.